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The low dimensional homology groups of the elementary group of degree two

Behrooz Mirzaii, Elvis Torres Pérez

TL;DR

The paper analyzes the low-dimensional homology of the elementary group $E_2(A)$ for a commutative ring $A$ by constructing and exploiting a first-quadrant spectral sequence derived from the unimodular-vector complex $Y_\bullet(A^2)$. It provides concrete descriptions of $H_1(E_2(A),\mathbb{Z})$ in terms of $A/M$, derives a fundamental exact sequence linking $H_2(E_2(A),\mathbb{Z})$ to the square-ideal $I^2(A)$, and identifies generators via explicit cycles, including Steinberg symbols. The core contribution is a refined Bloch–Wigner-type exact sequence for $H_3(E_2(A),\mathbb{Z})$ over semilocal rings, relating it to the refined Bloch group $\mathcal{RB}(A)$ and torus-related terms, under natural hypotheses on residue fields and unit groups. These results advance understanding of the third homology via projective refinements and illuminate connections to scissors-congruence groups, $K_2$-theory, and algebraic $K$-theory for low-dimensional homology.

Abstract

In this article we study the first, the second and the third homology groups of the elementary group $\textrm{E}_2(A)$, where $A$ is a commutative ring. In particular, we prove a refined Bloch-Wigner type exact sequence over a semilocal ring (with some mild restriction on its residue fields) such that $-1\in (A^{\times})^2$ or $|A^{\times}/(A^{\times})^2|\leq 4$.

The low dimensional homology groups of the elementary group of degree two

TL;DR

The paper analyzes the low-dimensional homology of the elementary group for a commutative ring by constructing and exploiting a first-quadrant spectral sequence derived from the unimodular-vector complex . It provides concrete descriptions of in terms of , derives a fundamental exact sequence linking to the square-ideal , and identifies generators via explicit cycles, including Steinberg symbols. The core contribution is a refined Bloch–Wigner-type exact sequence for over semilocal rings, relating it to the refined Bloch group and torus-related terms, under natural hypotheses on residue fields and unit groups. These results advance understanding of the third homology via projective refinements and illuminate connections to scissors-congruence groups, -theory, and algebraic -theory for low-dimensional homology.

Abstract

In this article we study the first, the second and the third homology groups of the elementary group , where is a commutative ring. In particular, we prove a refined Bloch-Wigner type exact sequence over a semilocal ring (with some mild restriction on its residue fields) such that or .
Paper Structure (6 sections, 16 theorems, 153 equations)

This paper contains 6 sections, 16 theorems, 153 equations.

Table of Contents

  1. Elementary and special linear groups of degree two
  2. The complex of unimodular vectors
  3. The main spectral sequence and the homology of Lg
  4. The first homology of Lg
  5. The second homology of Lg
  6. The third homology of Lg

Key Result

Theorem 1.1

Let $A=\mathbb{Z}[\sqrt{-d}]$ or $A=\mathbb{Z}[\frac{1}{2}(1+\sqrt{1-4d})]$, where $d$ is a positive integer. (i) If $d\geq 4$, then $\mathit{{\rm E}}_2(A)$ is a non-normal, infinite-index subgroup of $\mathit{{\rm SL}}_2(A)$. (ii) If $1\leq d < 4$, then $\mathit{{\rm E}}_2(A)=\mathit{{\rm SL}}_2(A)

Theorems & Definitions (39)

  • Theorem 1.1
  • proof
  • Theorem 1.2: Liehl liehl1981
  • proof
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.3: Hutchinson
  • proof
  • ...and 29 more